How Does a Spinning Rope Exert Force on Its Segments?

[stunner5000pt]

ASsume the z axis is the vertical axis

A homogenous rope of mass M and length L hangs vertically from a point of hte ceiling.
Find the magnitude of the force F(z) by which the top part of the rope acts on the lower segment of the rop at the distance z from the free end of the rope.

Not quite sure here Would i have to use something like mass density of each segment like how you do for a line of charge calculation??
so then define $$\lambda = \frac{M}{L}$$

so then $$F(z) = m_{eff} g$$
$$F(z) = \lambda z g = \frac{Mg}{L} z$$
good so far??

Now the rope is spun in a horizontal circle with constant angular velocity omega. Ignore gravity.
Find the magnitude of the force F(r) by which the inner part of the rope acts on the outer segment of the rope at the distance r from the center of the circular motion.
so use the same lambda as before
The question above has been copied from the book verbatim. I'm wondering if the rope is being spun about its center or from one end?
$$F(r) = m_[eff} \omega^2 r$$
$$F(r} = \lambda r \omega^2 r = \frac{M}{L} \omega^2 r^2$$

now i know this is wrong because th book gives the answer to be $$F(r) = \frac{M \omega^2}{2L} (L^2 - r^2)$$
where di i go wrong??

 

[ramollari]

Why did you take r as the radius? It is like there's a point mass of mass $$m_{eff}$$ moving in a circle of radius r. Certainly this is not correct. Divide the rope into tiny imaginary points of mass dm, find the force applied on each individual rope segment as $$dF = \omega ^2r_{pt} dm$$. Try to express r_{pt} in terms of the mass that remains from the extreme, total mass, and length L. Then integrate from 0 (mass supported at the extreme) to m that is the mass supported at the point where r_{pt} = r.

 

[thepatientmental]

Your calculations for the first part are correct. The force exerted by the top part of the rope on the lower segment at a distance z from the free end is F(z) = Mg/L * z. This is because the mass of the rope segment above z is Mz/L, and the force of gravity on that segment is Mgz/L.

For the second part, the rope is being spun about its center, so the force F(r) is the force exerted by the inner part of the rope on the outer segment at a distance r from the center of the circular motion. Your mistake is in using the same lambda as before. In this case, the mass of the rope segment above r is M(r/L), so the force of gravity on that segment is Mg(r/L).

To find the force F(r), we need to consider the centripetal force required to keep the rope in circular motion. This is given by F(r) = m_eff * \omega^2 * r, where m_eff is the effective mass of the rope segment above r, taking into account the centripetal force.

To find m_eff, we can use the concept of angular velocity, which is defined as \omega = v/r, where v is the linear velocity of the rope segment at a distance r from the center. We can also use the fact that the linear velocity is related to the angular velocity by v = \omega * r.

Substituting these relationships into the expression for centripetal force, we get F(r) = m_eff * \omega^2 * r = (M(r/L)) * (\omega^2 * r) = (M/L) * (r * \omega^2).

Therefore, the magnitude of the force F(r) is given by F(r) = (M/L) * (r * \omega^2) = (M * \omega^2)/L * (r * \omega^2) = (M * \omega^2)/(2L) * (r^2 - L^2).

Your mistake was in using the same lambda as before, which does not account for the centripetal force required to keep the rope in circular motion. By using the concept of angular velocity, we can correctly calculate the force F(r) exerted by the inner part of the rope on the outer segment.